Mathematical Concepts in Exoplanet Detection

This note explains the key mathematical concepts used in detecting exoplanets from light curves, with intuitive explanations, examples, and analogies to help understand these complex ideas.

1. Light Curves and Flux Measurement

Basic Concept

Think of a star as a steady light bulb. When a planet passes in front of it (like a small bug crawling across your flashlight), it temporarily blocks some light.

Key Formula

$$F(t) = \text{photons collected per unit time}$$

Normalization

We normalize the flux so that a quiet star sits at $F = 1.0$, making it easier to see small changes.

Example

If a star normally has a brightness of 1.0, and we observe it at 0.99, that means the star is 1% fainter than usual.

Analogy

Think of this like measuring the brightness of your car's headlights. On a clear night they might register 100 units, but if a bug flies across them briefly, they might drop to 99 units - that's a 1% dip.

2. Transit Depth - How Much Light is Blocked?

Key Formula

$$\delta = \frac{\Delta F}{F} = \left(\frac{R_p}{R_\star}\right)^{2}$$

Where:

• $\delta$ = transit depth (fractional drop in brightness)
• $R_p$ = planet radius
• $R_\star$ = star radius

Explanation

The amount of light blocked depends on the relative sizes of the planet and star - it's simply the ratio of their areas.

Examples

1. Jupiter-size planet:

• $R_p \approx 0.1\,R_\odot$
• $\delta \approx (0.1)^2 = 0.01$ → a 1% dip (~10,000 ppm)

1. Earth-size planet:

• $\delta \approx 84\text{ ppm}$ → 0.0084%
• This is incredibly small - like trying to detect a flea crawling across a car headlight from miles away!

Analogy

Imagine trying to measure how much sunlight is blocked when:

• A basketball passes in front of a basketball court floodlight (Jupiter-sized planet)
• A marble passes in front of the same floodlight (Earth-sized planet)

The marble would block almost no light at all!

3. The Three Key Transit Parameters

Every transit can be described by three observable quantities:

Depth ($\delta$)

• What it tells us: Planet size relative to star
• What sets it: $(R_p/R_\star)^2$
• Example: A 1% dip suggests a planet about 10% the size of its star

Duration ($T_{14}$)

• What it tells us: How long the transit lasts
• What sets it: Orbital distance and how centrally the planet crosses
• Formula: $$T_{14} \approx \frac{P}{\pi}\,\frac{R_\star}{a}\,\sqrt{1-b^{2}}$$
• Where $b$ is the impact parameter (how close to the center the planet crosses)

Period ($P$)

• What it tells us: How often the planet orbits its star (length of "year")
• What sets it: Kepler's third law
• Formula: $$P^{2} = \frac{4\pi^{2}}{G M_\star}\,a^{3}$$

Analogy

Think of a train passing in front of a streetlight:

• Depth: How much the light dims (depends on train size vs. light size)
• Duration: How long the light is dim (depends on train speed and how directly it passes)
• Period: How often the train returns (depends on track length and train speed)

4. Kepler's Third Law - Connecting Period to Distance

Formula

$$P^{2} = \frac{4\pi^{2}}{G M_\star}\,a^{3}$$

What it means

The square of a planet's orbital period is proportional to the cube of its average distance from the star.

Example

If a planet has an orbital period of 1 year around a Sun-like star, it's about 1 AU (astronomical unit) away. If its period is 8 years, it's about 4 AU away (4³ = 64, √64 ≈ 8).

Analogy

Think of cars on a circular racetrack:

• The farther a car is from the center (larger orbit), the longer it takes to complete one lap
• A car twice as far from the center takes about 2.8 times longer to complete a lap

5. Transit Duration - How Long Does the Eclipse Last?

Formula

$$T_{14} \approx \frac{P}{\pi}\,\frac{R_\star}{a}\,\sqrt{1-b^{2}}$$

Explanation

This tells us how long a transit lasts, which depends on:

• The orbital period ($P$)
• The star's radius relative to the orbital distance ($R_\star/a$)
• How centrally the planet crosses ($b$, the impact parameter)

Examples

1. Central transit ($b \approx 0$): Longest duration, flattest transit shape
2. Grazing transit ($b \to 1$): Shortest duration, V-shaped transit

Analogy

Think of a bird flying across the sun:

• If it flies directly across the center, it takes longer to cross and blocks the sun more completely
• If it just grazes the edge, it quickly passes by and blocks less light

6. Signal-to-Noise Ratio (SNR) - How Detectable is the Signal?

Formula

$$\text{SNR} \approx \frac{\delta}{\sigma}\,\sqrt{N_{tr}\,n_{\text{in}}}$$

Where:

• $\delta$ = transit depth
• $\sigma$ = per-point scatter (noise level)
• $N_{tr}$ = number of observed transits
• $n_{\text{in}}$ = points per transit

What it means

This tells us how confidently we can detect a transit above the noise.

Detection Threshold

Below SNR ≈ 7, detection is unreliable - this is the "detection floor".

Example

If we observe a planet that causes a 0.01 (1%) dip, but our measurements have noise of 0.005 (0.5%), we need to observe multiple transits to build confidence in our detection.

Analogy

Trying to hear someone whispering in a noisy room:

• The whisper is the signal (transit)
• The room noise is the background noise
• The more times you hear the same whisper, the more confident you are it's real and not just noise

7. Transit Shape Signatures - How to Tell What's Causing the Dip

Different astrophysical phenomena produce different shaped dips:

Planet Transit

• Shape: Shallow, flat-bottomed U-shape
• Depth: Small and physical (< 3%)
• Secondary eclipse: None (planets don't emit significant light)
• Odd/even transits: Match exactly

Eclipsing Binary

• Shape: Often V-shaped
• Secondary eclipse: Present (half a period later)
• Odd/even depth: Unequal (stars have different brightness)
• Depth: Often too deep for a planet

Blend/Contamination

• Diluted depth: Deep eclipse from neighbor star made to look shallow
• Centroid shift: Light source appears to move during the dip

Analogy

Think of different objects passing in front of a flashlight:

• A small ball bearing (planet): Makes a small, flat shadow that's consistent
• Two similarly sized balls (binary stars): Makes a large V-shaped shadow with a second shadow later
• A large ball partially blocked by a fence (blend): Makes a small shadow that seems to move